A *total domatic $k$-partition* of a graph is a partition of its vertex set into $k$ subsets such that each intersects the open neighborhood of every vertex.
The maximum $k$ for which a total domatic $k$-partition exists is known as the *total domatic number* of a graph $G$, denoted by $d_t(G)$.
We extend considerably the known hardness results by showing it is NP-complete to decide whether $d_t(G) \geq 3$ where $G$ is a bipartite planar graph of bounded maximum degree.
Similarly, for every $k \geq 3$, it is NP-complete to decide whether $d_t(G) \geq k$, where $G$ is a split graph or $k$-regular.
In particular, these results complement very recent combinatorial results regarding $d_t(G)$ on some of these graph classes by showing that the known results are, in a sense, best possible.
Moreover, for general $n$-vertex graphs, we show the problem is solvable in $3^n n^{O(1)}$ time and polynomial space, and derive even faster algorithms for special graph classes via simple reductions.
Finally, we briefly discuss the possibility of breaking the $3^n$ time barrier in polyspace.
In particular, this bound has been bypassed for the related problem of *domatic number*, but it appears plausible different ideas are needed for total domatic number.